Finite Group Databasebeta

← #27437754

Subgroups — #27437754

Order 13568.

1116 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811018C211yesmaximal, characteristic, complemented
2C2×#16810975C211yesmaximal, characteristic, complemented
3#20280481C211yesmaximal, characteristic, complemented
4#20275723C212yesmaximal, complemented
5#20275718C212yesmaximal
6#20275689C211yesmaximal, characteristic, complemented
7#20275527C211yesmaximal, characteristic, complemented
8#20275056C212yesmaximal, complemented
9#20275049C212yesmaximal, complemented
10#20274481C211yesmaximal, characteristic, complemented
11C2×C53×#542C211yesmaximal, characteristic
12C22×#11719925C2212yes—
13C22×#11719923C2212yes—
14C2×#13133217C2211yescharacteristic
15C2×#13133196C2212yes—
16#16811018—24no—
17#16810975—24no—
18C2×#13133141C2212yes—
19C2×#13133140C2212yes—
20C2×#13133140C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2212yes—
23C2×#13133128C2211yescharacteristic
24C2×C4×#88322C2211yescharacteristic
25#16810553—21no—
26#16810553—21no—
27#16810548—21no—
28#16810548—21no—
29#16810529C2212yes—
30#16810489C2211yescharacteristic
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#542—24no—
35C53×#542—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#126×C53—21no—
38C2×#124×C53—21no—
39C2×#124×C53C2211yescharacteristic
40C2×#123×C53—21no—
41C2×C22⋊Q8×C53—21no—
42C2×C22⋊Q8×C53C2211yescharacteristic
43C2×#121×C53C2211yescharacteristic
44C2×C4×Q8×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C2×#117×C53C2211yescharacteristic
47C22×Dic106C2312yes—
48C2×#11719925—24no—
49C2×#11719925—24no—
50C2×#11719925—42no—
51C2×#11719923—24no—
52C2×#11719923—24no—
53C2×#11719923—42no—
54C22×#88322C2312yes—
55C22×#88322C2312yes—
56#13133217—22no—
57#13133217—22no—
58#13133196—42no—
59#13133196—42no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53C2312yes—
63C2×#11719919—22no—
64C2×C22⋊Dic53—22no—
65#13133141—42no—
66#13133140—24no—
67#13133140—24no—
68#13133138—42no—
69C22⋊#88322—24no—
70#13133128—22no—
71C4×#88322—22no—
72#13133124—22no—
73#13133124—22no—
74#13133124—21no—
75#13133124—21no—
76C22×Q8×C53—21no—
77C22×Q8×C53C2311yescharacteristic
78C22×D4×C53—21no—
79C22×D4×C53C2311yescharacteristic
80C23×C4×C53—21no—
81C23×C4×C53C2311yescharacteristic
82C4⋊Q8×C53—22no—
83C4⋊Q8×C53D414yes—
84#126×C53—28no—
85#124×C53—28no—
86#124×C53—22no—
87#124×C53D414yes—
88#123×C53—28no—
89C22⋊Q8×C53—28no—
90C22⋊Q8×C53—22no—
91C22⋊Q8×C53—22no—
92#121×C53—22no—
93#121×C53—22no—
94C4×Q8×C53—22no—
95C4×Q8×C53—22no—
96C4×D4×C53—22no—
97C4×D4×C53—22no—
98#117×C53—22no—
99#117×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C24×C53—21no—
118C24×C53C2×D411yescharacteristic
119C2×Q8×C53—24no—
120C2×Q8×C53—21no—
121C2×Q8×C53—22no—
122C2×Q8×C53—22no—
123C2×Q8×C53—82no—
124C2×Q8×C53C2×D412yes—
125C2×D4×C53—24no—
126C2×D4×C53—21no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—82no—
130C2×D4×C53C2×D412yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—24no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—82no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—22no—
147C22×Dic53—22no—
148C2×Dic106—22no—
149C2×Dic106—22no—
150C2×Dic106—22no—
151C2×Dic106—22no—
152C2×Dic106—22no—
153C2×Dic106—22no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53C2×D412yes—
162C4⋊C4×C53C2×D412yes—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C42×C53—24no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53C2×D412yes—
174C42×C53C2×D412yes—
175C22⋊Dic53—44no—
176#11719925—48no—
177#11719923—48no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—42no—
181C4⋊Dic53—42no—
182#11719919—44no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322#4114yes—
189C23×C53—21no—
190C23×C53—22no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53C22×D411yescharacteristic
200Q8×C53—24no—
201Q8×C53—42no—
202Q8×C53—84no—
203D4×C53—24no—
204D4×C53—42no—
205D4×C53—84no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—84no—
225C2×C4×C53#12012yes—
226C2×C4×C53#12012yes—
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4112yes—
230C2×C4×C53H(5,𝔽2)12yes—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237Dic106—42no—
238Dic106—42no—
239Dic106—42no—
240Dic106—42no—
241#88322—24no—
242#88322—42no—
243#25613—531nomaximal, Sylow
244C22×C53—42no—
245C22×C53—41no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—42no—
252C22×C53—82no—
253C22×C53—82no—
254C22×C53C2×#12011yescharacteristic
255C22×C53#53211yescharacteristic
256C22×C53#53511yescharacteristic
257C22×C53#57712yes—
258C22×C53#57812yes—
259C4×C53—22no—
260C4×C53—22no—
261C4×C53—41no—
262C4×C53—42no—
263C4×C53—41no—
264C4×C53—42no—
265C4×C53—42no—
266C4×C53—41no—
267C4×C53#54612yes—
268C4×C53#55012yes—
269Dic53—82no—
270Dic53—82no—
271C2×#542D5311yescharacteristic
272C2×#497—531no—
273C2×#487—531no—
274#2621—531no—
275#2001—532no—
276#2000—532no—
277#1993—531no—
278#1958—531no—
279#1878—532no—
280#1877—532no—
281#1750—531no—
282C2×C53—81no—
283C2×C53—82no—
284C2×C53#243111yescharacteristic
285C2×C53#300911yescharacteristic
286C2×C53#301111yescharacteristic
287C2×C53#317812yes—
288C2×C53#322512yes—
289C22×Q16—532no—
290C22×SD16—532no—
291#542—24no—
292#542—24no—
293C2×C4⋊Q8—531no—
294C2×C4⋊Q8C2×D5311yescharacteristic
295C2×#126—21no—
296C2×#124—21no—
297C2×#124C2×D5311yescharacteristic
298C2×#123—21no—
299C2×C22⋊Q8—21no—
300C2×C22⋊Q8—532no—
301C2×C22⋊Q8C2×D5311yescharacteristic
302C2×#121C2×D5311yescharacteristic
303C2×C4×Q8C2×D5311yescharacteristic
304C2×C4×D4C2×D5311yescharacteristic
305C2×#117C2×D5311yescharacteristic
306#497—1064no—
307#487—1064no—
308C2×#103—532no—
309C2×#103—532no—
310C2×#103—532no—
311C2×#102—532no—
312C2×C22⋊C8—532no—
313C2×#97—531no—
314C2×C4×C8—531no—
315#398—1061no—
316#398—1061no—
317#397—1061no—
318#397—1061no—
319#391—532no—
320#382—531no—
321#340—532no—
322#340—532no—
323#336—532no—
324C53#2561311yescharacteristic, complemented, Sylow
325C22×Q8—21no—
326C22×Q8—532no—
327C22×Q8C22×D5311yescharacteristic
328C22×D4—21no—
329C22×D4C22×D5311yescharacteristic
330C23×C4—21no—
331C23×C4C22×D5311yescharacteristic
332C2×Q16—1064no—
333C2×Q16—1064no—
334C2×Q16—2122no—
335C2×SD16—1064no—
336C2×SD16—1064no—
337C2×SD16—2122no—
338C22×C8—532no—
339C22×C8—532no—
340C4⋊Q8—22no—
341C4⋊Q8—1062no—
342C4⋊Q8—1062no—
343C4⋊Q8C22⋊D5314yes—
344#126—28no—
345#124—28no—
346#124—22no—
347#124C22⋊D5314yes—
348#123—28no—
349C22⋊Q8—28no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—2122no—
353C22⋊Q8—2122no—
354#121—22no—
355#121—22no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×D4—22no—
359C4×D4—22no—
360#117—22no—
361#117—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—532no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4C22⋊D5311yescharacteristic
370C2×C4⋊C4C22⋊D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4C22⋊D5311yescharacteristic
380C2×C22⋊C4C22⋊D5311yescharacteristic
381C2×C42—21no—
382C2×C42C22×D5311yescharacteristic
383C2×C42C22×D5311yescharacteristic
384#103—1064no—
385#103—1064no—
386#103—2122no—
387#102—2122no—
388C22⋊C8—1064no—
389#97—1062no—
390C4×C8—1062no—
391#95—1061no—
392#95—1061no—
393#95—1062no—
394#95—1062no—
395C24—21no—
396C24C2×C22⋊D5311yescharacteristic
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—82no—
402C2×Q8—1062no—
403C2×Q8—1062no—
404C2×Q8—1062no—
405C2×Q8—1062no—
406C2×Q8—1062no—
407C2×Q8—1062no—
408C2×Q8C2×C22⋊D5312yes—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—82no—
414C2×D4C2×C22⋊D5312yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—82no—
424C22×C4—1062no—
425C22×C4—1062no—
426C22×C4C2×C22⋊D5311yescharacteristic
427C22×C4C2×C22⋊D5311yescharacteristic
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C23×D5311yescharacteristic
432Q16—2128no—
433SD16—2128no—
434C2×C8—534no—
435C2×C8—1062no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C2×C8—1062no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—1062no—
448C4⋊C4—1062no—
449C4⋊C4—2124no—
450C4⋊C4—2122no—
451C4⋊C4—2122no—
452C4⋊C4C2×C22⋊D5312yes—
453C4⋊C4C2×C22⋊D5312yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—2124no—
462C42—24no—
463C42—21no—
464C42—21no—
465C42C2×C22⋊D5312yes—
466C42C2×C22⋊D5312yes—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—22no—
471C23—22no—
472C23—21no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23C22×C22⋊D5311yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—84no—
481Q8—2122no—
482Q8—2122no—
483Q8—2122no—
484Q8—2122no—
485D4—24no—
486D4—42no—
487D4—84no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4—2122no—
511C2×C4—2122no—
512C2×C4—2122no—
513C2×C4#1313318612yes—
514C2×C4#1313318612yes—
515C2×C4C22×C22⋊D5311yescharacteristic
516C2×C4C22×C22⋊D5311yescharacteristic
517C2×C4#41×D5312yes—
518C2×C4#1313326312yes—
519C8—1064no—
520C8—2122no—
521C22—41no—
522C22—42no—
523C22—42no—
524C22—41no—
525C22—41no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—82no—
530C22—82no—
531C22C2×#1313318611yescharacteristic
532C22#1681116911yescharacteristic
533C22#1681118711yescharacteristic
534C22#1681143412yes—
535C22#1681143812yes—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—42no—
540C4—41no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—4242no—
545C4—4242no—
546C4#1681126112yes—
547C4#1681129212yes—
548C2—81no—
549C2—82no—
550C2#2027833011yescharacteristic
551C2#2028399211yescharacteristic
552C2#2028401011yescharacteristic
553C2#2028537712yes—
554C2#2028581812yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#542C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#542D53
hypocenterC53#25613
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16811018C2semidirect
2. C2×#16810975C2semidirect
3. #20280481C2semidirect
4. #20275723C2semidirect
6. #20275689C2semidirect
7. #20275527C2semidirect
8. #20275056C2semidirect
9. #20275049C2semidirect
10. #20274481C2semidirect
324. C53#25613semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168110182
C2×#168109752
#202804812
#202757232
#202757182
#202756892
#202755272
#202750562
#202750492
#202744812
C2×C53×#5422
#2561353